To solve this problem, we can use trigonometry, specifically the tangent function, because we have a right triangle formed by the height of the balloon, the horizontal distance between the station and the balloon, and the line of sight from the station to the balloon.

Given:

• Distance between the ground crew and the observation station: $1.2 \, \text{miles}$
• Angle of elevation: $30^\circ$
• We are looking for the height $h$ of the balloon.

The tangent of an angle in a right triangle is the ratio of the opposite side (height) to the adjacent side (horizontal distance). Mathematically:

$\tan(\theta) = \frac{\text{height}}{\text{distance}}$

Rearranging to solve for height:

$h = \tan(\theta) \times \text{distance}$

Substitute the known values:

$h = \tan(30^\circ) \times 1.2 \, \text{miles}$

We know that $\tan(30^\circ) \approx 0.577$.

So,

$h = 0.577 \times 1.2 \, \text{miles}$

$h \approx 0.6924 \, \text{miles}$

Final Answer:

The height of the balloon is approximately 0.692 miles. To convert this into feet, multiply by 5280 feet per mile:

$0.692 \, \text{miles} \times 5280 \, \text{feet/mile} \approx 3651.36 \, \text{feet}$

Thus, the balloon's height is approximately 3651 feet.

Would you like to see a detailed explanation of the steps or have any further questions?

Here are 5 related questions:

1. How would the answer change if the angle of elevation were different?
2. What other trigonometric functions could be used in this scenario?
3. How would the height be affected if the distance between the crew and the observation station was increased?
4. How would you apply this method if the balloon were moving horizontally as well?
5. What would the calculation be if the angle of elevation was 45 degrees instead?

Tip: Always make sure to convert angles into the correct units (degrees or radians) based on the calculator you are using.